Formations of satellites in co-planar orbits around a single body are increasingly being used to perform functions that a single spacecraft would be unable to do. To optimize the formation, the inter-satellite spacing should be maintained over the lifetime of the vehicles. This requires the satellites to achieve the desired inter-satellite separation after launch, to compensate for any perturbing forces they encounter while in orbit and to redistribute the formation if one or more satellites experience a systemic failure. In the past, formation assembly and maintenance have been performed using expendable-mass propulsion systems. However, many spacecraft, particularly micro- and nano-satellites, have extreme space and mass constraints which make it difficult to incorporate a propulsion system that depends on expendable mass. Expendable-mass propulsion systems also increase the cost and complexity of the spacecraft. It is therefore desirable and cost effective to use naturally occurring forces to perform formation establishment and maintenance.
Atmospheric drag is typically a dominant perturbing force for spacecraft in Low Earth Orbit (“LEO”). Differential atmospheric drag can arise due to variations in atmospheric density, variations in temperature, differences in satellite mass profiles, and variations in spacecraft drag coefficients, among others. The acceleration due to atmospheric drag is given by the following equation:
                    a        =                              1                          2              ⁢              M                                ⁢          ρ          ⁢                                          ⁢                      V            2                    ⁢                      AC            D                                              (        1        )            
M is the spacecraft mass, p is the atmospheric density, V is the velocity, A is the cross-sectional area in the direction of motion. CD is the drag coefficient, and acceleration acts in the negative velocity direction. While the mass and drag coefficient of the spacecraft cannot be easily changed during flight, the cross-sectional area may be varied with minimal effort.
Several studies have advocated the use of differential drag to maintain spacing between multiple satellites or to perform rendezvous maneuvers between multiple spacecraft. In these studies, several approaches to the problem of altering the cross-sectional area have been proposed. C. L. Leonard suggested varying the angle of attack of a plate on the spacecraft in order to vary the cross-sectional area. See, e.g., C. L. Leonard, “Formation keeping of Spacecraft via Differential Drag, M. S. Thesis, Massachusetts Institute of Technology, July 1986, incorporated herein by reference. Additional solutions include drag plates which can be opened or closed, solar panel orientation, and orienting the spacecraft to achieve the desired cross-sectional area. See, e.g., U.S. Pat. No. 5,806,801; Balaji Shankar Kumar, Alfred Ng, Keisuke Yoshihara and Anion De Ruiter. “Differential Drag as a means of Spacecraft Formation Control.” Proceedings of the IEEE Aerospace Conference, Big Sky, Mont., March 2007: Balaji Shankar Kumar and Alfred Ng, “A Bang-Bang Control Approach to Maneuver Spacecraft in a Formation With Differential Drag,” AIAA Guidance, Navigation and Control Conference and Exhibit, Aug. 18-21, 2008, Honolulu, Hi., AIAA 2008-6469; Balaji Shankar Kumar, Alfred Ng, and Keisuke Yoshihara “Flight Dynamics and Control of the JC2Sat Mission,” Advances in the Astronautical Sciences, Vol 129, Pert 3, 2008 (AAS 07-410); Riccardo Bevilacqua and Marcello Romano, “Rendezvous Maneuvers of Multiple Spacecraft Using Differential Drag Under 12 Perturbation,” Journal of Guidance. Control and Dynamics, Vol. 3 1. No. 6, November-December 2008; Timothy Maclay and Christopher Tuttle, “Satellite Station keeping of the ORBCOMM Constellation Via Active Control of Atmospheric Drag: Operations, Constraints, and Performance,” Advances in the Astronautical Sciences, Vol. 120, Part I, pp. 763-773, all incorporated by reference.
The use of differential drag for formation maintenance has been successfully demonstrated by OrbComm Inc. OrbComm satellites rely on solar panel orientation to increase the cross-sectional area in the desired direction (i.e., either the direction of motion, or the direction of solar radiation pressure) while the satellite is in eclipse or has excess power while in sunlight. At other times, the attitude of the spacecraft is adjusted to increase or decrease the projected area in a desired direction. Attitude control is an effective method for increasing the cross-sectional area only when the spacecraft uses symmetric antenna patterns. U.S. Pat. No. 5,806,801, which discusses the method and system employed by the OrbComm satellites, discloses that the radiation patterns used by antenna subsystems are substantially symmetrical such that radio transmissions are not adversely affected by a yaw angle induced to increase the cross-sectional area in the desired direction. Satellites with asymmetric antenna patterns must remain directionally-fixed in the Local Vertical. Local Horizontal (“LVLH”) system and as such cannot use attitude control as a means to increase cross-sectional area.
In previous studies, the focus has been on small-scale maneuvers such as formation maintenance and/or satellite maneuvering in close proximity. In the case of the OrbComm constellation, formation assembly is mentioned, but it is specified that propellant-fueled thrusters will be used to help move the spacecraft closer to the desired final trajectory before differential drag is employed to make smaller adjustments to the orbit profile. Mathews and Leszkiewicz developed methods to control relative decay between two satellites and therefore minimize fuel required for formation keeping. See, e.g., Michael Mathews and Susan Leszkiewicz, “Efficient Spacecraft Formationkeeping With Consideration of Ballistic Coefficient Control,” AIAA 26th Aerospace Sciences Meeting, AA-88-0375, January 1988, incorporated herein by reference. C. I. Leonard developed control laws to maintain close-proximity spacing between two vehicles. Kumar et al similarly developed a control method to maintain spacing between two satellites and to perform maneuvers to vary the separation distance between 20 km and 100 m. In these three cases, the control methods were developed using the Hill-Clohessy-Wiltshire (HCW) equations. The maneuvers proposed by Bevilacqua and Ramano are, by definition, close-proximity maneuvers and are based on HCW equations that have been modified to include the J2 effect. The J2 effect relates to a gravitational perturbations model that partially accounts for the Earth not being perfectly spherical. See, e.g., Riccardo Bevilacqua and Marcello Romano, “Rendezvous Maneuvers of Multiple Spacecraft Using Differential Drag Under 12 Perturbation,” Journal of Guidance, Control and Dynamics, Vol. 3 1, No. 6, November-December 2008, incorporated herein by reference.